1. Suppose we consider some function on the domain . For example: We revolve this about the axis by 360 degrees. So this looks like Question: what is the area occupied by this surface? Advertisements
1. Try integrating (1)??? It’s kind of hard, so we try to do something very clever: use the chain rule.
So recall we introduced the natural logarithm. Integrals interpret the natural logarithm as , and we will explore that in this post.
1. Introduction. There are about 4 properties we expect the integral to obey. We expect an integral over a zero width region should vanish (1) The fundamental theorem of calculus confirms this. If is the antiderivative for , we see … Continue reading
1. Review. So we introduced some technique to figure out the area between a curve and the line . This was mildly complicated involving sums, partitions, and other exotic names. The question: can’t we do something simpler? The answer: yes…kind … Continue reading
1. Estimating Area of Shapes. We can consider limits of finite sums when estimating the areas of shapes in the plane. So for example, the triangle: Note that (1) describes the triangle’s hypoteneuse. What can we do? Well, we can … Continue reading